Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power Networks

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Luis Miguel Castro González

Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power Networks

Julkaisu 1445 • Publication 1445

Tampere 2016

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Tampereen teknillinen yliopisto. Julkaisu 1445 Tampere University of Technology. Publication 1445

Luis Miguel Castro González

Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power

Networks

Thesis for the degree of Doctor of Science in Technology to be presented with due permission for public examination and criticism in Sähkötalo Building, Auditorium S2, at Tampere University of Technology, on the 5^th of December 2016, at 12 noon.

Tampereen teknillinen yliopisto - Tampere University of Technology

Tampere 2016

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ISBN 978-952-15-3866-7 (printed)

ISBN 978-952-15-3869-8 (PDF)

ISSN 1459-2045

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iii

Abstract

Power transmission systems are expanded in order to supply electrical energy to remote users, to strengthen their operational security and reliability and to be able to carry out commercial transactions with neighbouring power grids. Power networks successfully expand, at any time, by incorporating the latest technological developments. A case in point is the current growth of electrical power grids, in terms of both infrastructure and operational complexity, to meet an unprecedented upward trend in global demand for electricity. Today’s expansion of the power grid is being supported, more and more, by power electronics, in the form of flexible AC transmission systems and HVDC systems using vo ltage-sourced converters. The latter option, in particular, has been developing exceedingly rapidly since 1999, when the first commercial VSC-HVDC transmission installation was commissioned; the voltage-sourced converter technology has become the preferred option for transporting electricity when natural barriers are present (e.g. the sea), in Europe and further afield. At least one influential European body has recommended that the next step in the construction of the so-called European super grid, be a meshed HVDC transmission system based on the use of the VSC technology, to facilitate further the massive incorporation of renewable energy sources into the Pan European electrical grid, the harvesting of hydrocarbons that lie in deep sea waters and the energy trading between neighbouring countries.

It is in this context that this thesis contributes new knowledge to the modelling of VSC-based equipment and systems for the assessment of steady-state and transient stability analyses of AC/DC power networks. The STATCOM, the back-to-back VSC-HVDC link, the point-to-point VSC-HVDC link and the multi-terminal VSC-HVDC link, all receive research attention in this work.

The new models emanating from this research capture all the key steady-state and dynamic characteristics of the equipment and network. This has required a paradigm shift in which the VSC equipment has been modelled, here-to-fore, assuming that a voltage-sourced converter behaves like an idealised voltage source. In contrast, the models developed in this research resort to an array of basic power systems elements, such as a phase-shifting transformer and an equivalent shunt susceptance, giving rise to a two-port circuit where the AC and DC sides of the VSCs are explicitly represented. The ensuing VSC model is fundamentally different from the voltage source model; it represents, in an aggregated manner, the array of semiconductor switches in the converter and its PWM control. The VSC model was used as the basic building block with which to develop all the VSC-based devices put forward in this thesis. The ultimate device is the multi- terminal VSC-HVDC system, which may comprise an arbitrary number of VSC units commensurate with the number of otherwise independent AC sub-networks and a DC network of an arbitrary topology. The steady-state and dynamic simulations of the AC/DC systems are carried out using a unified frame-of-reference which is amenable to the Newton-Raphson algorithm. This framework accommodates, quite naturally, the set of discretised differential equations arising from the synchronous generators, HVDC and FACTS equipment, and the algebraic equations describing the conventional transmission lines, transformers and loads of the AC sub-networks.

The application areas covered in this work are: power-flow studies and dynamic simulations.

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Preface

Completion of this doctoral thesis was possible due to the help and support of several people. I would like to express my sincere gratitude to all of them.

The research work presented in this thesis was carried out at the Department of Electrical En- gineering (DEE), at Tampere University of Technology (TUT), to which I thank for having accepted me to be part of its vibrant academic atmosphere, first as a visiting student and then as a PhD student.

I am deeply grateful to Prof. Enrique Acha to whom I thank for supervising my thesis, providing me always with his enormous expertise and due guidance throughout my research. I would like to recognize that the years I spent in Finland under his supervision were the most fruitful in terms of my professional growth. I thank him for having facilitated my research in Finland with his uncondi- tional support and friendship. I would like to also thank Prof. Seppo Valkealahti for having accepted me a visiting student when he was the Head of Department of Electrical Engineering.

I am grateful to Prof. Ali Abur and Prof. Neville R. Watson for examining my manuscript and for providing me valuable comments which, undoubtedly, helped improve the quality of this thesis.

The s taff of the DEE deserves my regards for providing me with valuable assistance in practical matters of the workplace.

I thank all the c olleagues and friends that I encounter by chance in Tampere for their s upport and friendship, especially thanking Luis Enrique, Carlos, Jorge Paredes, Rodolfo, Angélica, Mafer, Moramay, Viri, Oscar, Kalle, Gabriel, Josmar, Victor, René, Rosa, Gaby, Pancho, Jan Servotte, Nemät Dehghani, Andrii and many others with whom I spent unforgettable moments.

Last but not least, I express my most profound gratitude to my parents, Luis and Guadalupe, and my siblings, Lupita, Victor, Leonardo and Cesar for their all-important encouraging words during the time I spent in Finland.

México City, November 2016 Luis Miguel Castro González

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List of Figures

Figure 3.1 (a)STATCOM schematic representation; (b) VSC steady-state equivalent circuit ... 16

Figure 3.2 Simplified representation of the power-flow model for the OLTC transformer; (a) Tap in nominal position, (b) Tap in off-nominal position... 19

Figure 3.3 VSC-HVDC sc hematic representation ... 23

Figure 3.4 Steady-state equivalent circuit of the VSC-HVDC link ... 25

Figure 3.5 Multi-terminal VSC-HVDC system interconnecting power networks of various kinds .. 35

Figure 3.6 Steady-state equivalent circuit of a three-terminal VSC-HVDC link ... 36

Figure 3.7 Flow diagram of a true unified solution of the multi-terminal VSC-HVDC system ... 47

Figure 4.1 Schematic diagram of the STATCOM and its control variables ... 51

Figure 4.2 DC voltage controller of the VSC ... 52

Figure 4.3 DC power controller for the DC side of the VSC... 53

Figure 4.4 AC-bus voltage controller of the VSC ... 53

Figure 4.5 Schematic representation of the VSC-HVDC link and its control variables ... 57

Figure 4.6 DC voltage dynamic controller of the VSC-HVDC link ... 58

Figure 4.7 DC-power controller of the VSC-HVDC link model ... 59

Figure 4.8 AC-bus voltage controllers: (a) rectifier station and (b) inverter station ... 60

Figure 4.9 Frequency controller of the VSC-HVDC link ... 67

Figure 4.10 Full VSC station with ancillary elements ... 71

Figure 4.11 Representation of a three-terminal VSC-HVDC link and its control variables ... 72

Figure 4.12 DC voltage dynamic controller of the slack converter VSC_Slack ... 73

Figure 4.13 (a) DC-power controller of the converter of type VSCP sch and (b) Frequency controller of the converter of type VSCPass ... 74

Figure 4.14 Modulation index controllers of the three converter stations making up the three- terminal HVDC system... 74

Figure 5.1 New England test system with two embedded STATCOMs ... 87

Figure 5.2 Angular speed of the synchronous generators ... 89

Figure 5.3 Active power flow behaviour in some transmission lines... 89

Figure 5.4 Reactive power flow behaviour in some transmission lines ... 90

Figure 5.5 Voltage performance at different nodes of the network ... 90

Figure 5.6 Reactive power generated by the STATCOMs... 91

Figure 5.7 Voltage performance at the AC nodes of the VSCs ... 91

Figure 5.8 Voltage performance at several nodes of the network including two STATCOMs ... 91

Figure 5.9 STATCOMs DC-bus voltages ... 92

Figure 5.10 STATCOMs DC current ... 92

Figure 5.11 Dynamic performance of the angle  of the VSCs ... 93

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Figure 5.12 Total active power losses incurred by the STATCOMs ... 93

Figure 5.13 Dynamic behaviour of the modulation index of the STATCOMs ... 94

Figure 5.14 Performance of the DC voltages for different ratings of the capacitors... 95

Figure 5.15 Performance of the modulation ratio for different ratings of the capacitors ... 95

Figure 5.16 Reactive power generation for different ratings of the capacitors ... 95

Figure 5.17 Test system used to validate the proposed VSC-HVDC model ... 96

Figure 5.18 DC voltage performance for the proposed and Simulink models ... 97

Figure 5.19 DC power performance for the proposed and Simulink models ... 98

Figure 5.20 Modulation indices performance for the proposed and Simulink models... 99

Figure 5.21 New England test system with embedded VSC-HVDC link ... 101

Figure 5.22 Voltage performance at different nodes of the network ... 102

Figure 5.23 Dynamic behaviour of the modulation indices ... 103

Figure 5.24 Reactive power generated by the VSCs of the HVDC link ... 103

Figure 5.25 DC current behaviour of the rectifier and inverter ... 104

Figure 5.26 DC voltage behaviour of the VSC-HVDC system ... 104

Figure 5.27 AC active powers and DC power behaviour of the HVDC link ... 105

Figure 5.28 Dynamic performance of the various angles involved in the HVDC link ... 105

Figure 5.29 Dynamic behaviour of the frequency at the inverter’s AC terminal and DC power: (a) Simulink model; (b) Proposed model ... 106

Figure 5.30 Dynamic performance of the DC voltages and modulation indices: (a) Simulink model; (b) Proposed model ... 107

Figure 5.31 VSC-HVDC link feeding into a low-inertia network ... 108

Figure 5.32 Voltage behaviour in the low-inertia network ... 109

Figure 5.33 Frequency behaviour in the low-inertia network ... 109

Figure 5.34 Dynamic performance of the AC and DC powers of the HVDC link... 110

Figure 5.35 Dynamic performance of _Rand I_dcI ... 110

Figure 5.36 Dynamic performance of the DC voltages and modulation indices... 111

Figure 5.37 Frequency in the low-inertia network and DC voltage of the inverter for different gains of the frequency control loop ... 112

Figure 5.38 Three-terminal VSC-HVDC link used to carry out the validation test ... 113

Figure 5.39 DC voltages of the VSCs comprising the three-terminal VSC-HVDC link. (a) Simulink model; (b) Developed model... 114

Figure 5.40 DC power behaviour at the DC bus of the three VSCs. (a) Simulink model, (b) Developed model ... 114

Figure 5.41 Modulation ratio of the three VSCs. (a) Simulink model; (b) Developed model... 115

Figure 5.42 Schematic representation of a multi-terminal VSC-HVDC link with a DC ring ... 116

Figure 5.43 Converter’s DC voltage and power flows in the DC grid ... 119

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ix Figure 5.44 Modulation ratio of the VSCs and frequency of the passive networks fed by VSC_b, VSCd, VSCf... 119

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List of Tables

Table 3.1 Types of VSCs and their control variables ... 38

Table 5.1 Parameters of the STATCOMs ... 87

Table 5.2 STATCOM results as furnished by the power-flow solution ... 88

Table 5.3 Initial values of the STATCOM variables for the dynamic simulation ... 88

Table 5.4 Parameters of the VSC-HVDC link ... 97

Table 5.5 VSC-HVDC va riables for the proposed model and Simulink model ... 99

Table 5.6 Parameters of the embedded VSC-HVDC link ... 100

Table 5.7 VSC-HVDC results given by the power-flow solution ... 101

Table 5.8 Initial values of the STATCOM variables for the dynamic simulation ... 102

Table 5.9 Parameters of the VSC-HVDC link with frequency regulation capabilities ... 106

Table 5.10 Parameters of the VSC-HVDC link feeding into a low-inertia network ... 108

Table 5.11 VSC-HVDC results given by the power-flow solution ... 109

Table 5.12 Different gain values for the frequency controller ... 111

Table 5.13 Parameters of the three-terminal VSC-HVDC link ... 113

Table 5.13 Parameters of the VSCs, AC1, AC3, AC5 and DC networks ... 116

Table 5.14 State variables solution for each VSC ... 117

Table 5.15 Voltages and injected powers at the AC networks ... 118

Table 5.16 Power flows in the DC ring... 118

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List of Symbols and Abbreviations

 _,V ^: Phase angle and magnitude of nodal voltages Pg,Qg: Generated active and reactive powers

Pd k_,Q_dk^: Active and reactive powers drawn by the load at bus k

cal

Pk _,Q_k^cal: Active and reactive power injections at bus k Beq: Equivalent shunt susceptance of the converter Cdc^: DC capacitor of the converter

Hc: Inertia constant of the converter’s DC capacitor Edc^: DC bus voltage

ma^: Modulation ratio of the converter

 : Phase-shifting angle of the converter

Gs w: Conductance for computing the switching losses of the converter ,

R X : Series resistance and reactance of the converter

ltc, ltc

R X ^: Series resistance and reactance of the OLTC transformer , ,

f f f

R X B : Resistance, reactance and susceptance of the AC filter of the converter

dc, dc

G R ^: Conductance and resistance of the DC links Inom^: Nominal current of the converter

Snom: Rated apparent power of the converter AC/DC: Alternating current/Direct current EMTS: Electromagnetic transient simulation IGBT: Insulated gate bipolar transistors

J : Jacobian matrix

HVDC: High-voltage direct current OLTC: On-load tap changer PWM: Pulse-width modulation

RMS: Root mean square

STATCOM: Static synchronous compensator

SVC: Static VAR compensator

VSC: Voltage source converter

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1 Introduction

Population growth and its migration to already large metropolitan areas, continue apace. To avoid the collapse of entire megalopolis in various parts of the world, the use of advanced technology has become essential. In particular, technology driven by electricity has enabled large number of people around the globe to reach higher levels of comfort, enabling more sophisticated human activities. All this is coming with a heavy price tag on the use of natural resources, gas emissions into the atmosphere beyond what it is deemed sustainable, toxic waste into the sea, deforestation an open mining of large tracts of land in the quest for metals and hydrocarbons. All this activity is clearly non-sustainable and modern society requires a paradigm shifts so that the new challenges are adequately resolved.

Concerning the remit of this work, electrical power networks have been subjected to a relentless expansion in order to meet the spiralling demand for electricity. This has brought great many challenges to the electricity supply industry as a whole. A case in point is the need to maintain the operational integrity of the system in the face of a large penetration of renewable energy sources and the widespread incorporation of power electronics equipment into the power grid to make it elec- tronically controllable. Most of the so-called renewable energy sources of electricity use primary energy resources that exhibit important degrees of randomness and use power electronics to make them amenable to their incorporation into the power grid. This has led to a more flexible electrical power grid but also one where the operational complexity has increased by a degree. All this comes at a time when large tracts of the electrical power grid hav e aged and are up for renewal, such as transmission lines, transformers, generators and protection equipment. All this infrastructure is critical to the well-being of an electrical power grid that would be “fit for purpose” to satisfy the users’ expectations for the delivery of high quality electricity at their points of connection.

The fulfilment of such high expectations would require the incorporation of cutting-edge technology and operating practises and tools. The aim would be to enable the supply of electrical energy at a reasonable cost, with low power losses in the distribution and bulk power transmission systems, bearing in mind reliability of supply and energy efficiency as the key drivers. In parallel with the

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renewal of the critical infrastructure, the development of state-of-the-art models of the emerging electrical equipment, control methods and simulation tools, requires continuous global effort.

These present and future developments would provide the theoretical foundations upon which the new, smarter power grids will be built. It is argued in this research work, that power systems simulation tools such as power flows and time-domain simulations would turn out to be the instrumental tools that will assist in the decision-making process at the design, planning and operation stages of future powe r grids.

The use of power electronic devices to enable a more flexible power grid aimed at achieving higher throughputs and enhancing system stability are high on the agenda in many countries around the globe; specifically, through the use of voltage source converters. The latest innovations in the power electronics field in terms of semiconductor valves, bridges and control strategies are motivating manufacturers to launch new converter designs into the global market, opening the gates to a flurry of new applications. With more power electronics than ever before embedded into the power system, a great deal of both technology and modelling issues are yet to be resolved. In the particular field of power converter modelling, for utility-scale power systems simulations, there has been a very creditable effort. However, it is often said that a voltage-sourced converter (VSC), including its derived family of equipment (e.g., VSC-HVDC), operates according to the basic prin- ciples of a controllable voltage source. It is argued here that this may be so only from a superficial standpoint, applicable only to power system studies of limited scope. A deeper analysis of this grid- connected equipment leads to the c onclusion that they are very rich in dynamics, whose calculation success cannot always be guaranteed, if the underrepresented idealised voltage source concept is used. Hence, one would need to resort to application tools which model the VSC at the in- dividual valve-level, such as PSCAD-EMTDC, the SimPowerSystems block set within Simulink or, alternatively, the new type of representations put forward in this research; each with their advantages and disadvantages.

This research work advances the topic of grid-connected VSC-based equipment modelling aimed at the steady-state and dynamic simulations of power systems. The modelling of STAT- COMs, back-to-back HVDC links, point-to-point HVDC links and multi-terminal HVDC systems are all addressed here. These models have been developed bearing in mind the key operational characteristics of the VSC-based equipment which will impact directly on the operation of the connected power grids. The encapsulation of the equipment’s key dynamic effects and efficient numerical solutions of the ensuing models in terms of accuracy and computing times for steady-state and dynamic analyses, have been issues of paramount importance. The family of models of high- voltage, high-power VSC-based equipment which are introduced in this thesis, signify a paradigm shift in the way this equipment has traditionally been modelled with respect to their fundamental frequency, positive-sequence representation of their AC circuits. It ought to be remarked that these models depart from the widely-adopted modelling practice of representing the VSC as idealised

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3 voltage or current sources. It is confirmed in this research work, using c redible “what if scenarios”

that these lumped models retain the key dynamics characteristics of the equipment by comparing their responses against the switching-based models available in the SimPowerSystems block set of Simulink. The STATCOM and the back-to-back, point-to-point and multi-terminal HVDC systems using VSCs , all received research attention.

1.1 Objectives and scientific contribution

The main objective of this thesis is to advance the fundamental-frequency, positive-sequence modelling, at their AC side, of the STATCOM, back-to-back HVDC link, point-to-point HVDC link and multi-terminal HVDC systems suitable for power systems simulations, with reference to their steady-state and dynamic operating regimes. Furthermore, a new frame-of-reference has been put forward in order to accommodate, in a unified manner, the new models and the positive-sequence model of the AC power network. The steady-state and dynamic models of the STATCOM, two- terminal VSC-HVDC links and multi-terminal VSC-HVDC systems put forward in this thesis represent the most comprehensive RMS-type models available today, to solve AC/DC power systems of realistic size. For instance, as far as this author is aware, there is no commercial package which contains RMS-type transient stability models that possess greater modelling flexibility than the models put forward in this timely piece of research. The same may be said about the models reported in the open literature, which neglect key dynamics and power losses of the VSC and VSC- HVDC equipment, such as switching losses, amplitude modulation indices, etc. Certainly, the dynamics of multi-terminal VSC-HVDC using RMS-type models do not seem to have been tackled anywhere.

In this research work, the AC circuit of the VSC and derived equipment are modelled using the positive-sequence framework and their representation depart from the customary idealised voltage source concept where key operational variables do not find representation. Furthermore, most of the adopted solution approaches found in the open literature are of sequential nature or, at best, quasi-Newton methods. This introduces frailty into their s olution approaches. In contrast, the new models take into account the converters’ current-dependent power losses, the explicit representation of the converters’ DC buses, the phase-shifting and scaling nature of the PWM control and the converters’ control strategies that fulfil their operating requirements to conform to the type of pairing AC s ub-network, an issue which becomes critical when applied to multi-terminal schemes. The set of algebraic and dynamic equations describing the new models and those of the whole network are solved in a truly unified manner using the Newton-Raphson method for solutions with quadratic convergence. In the case of time-domain solutions, the differential equations are discretised using the implicit trapezoidal method for enhanced numerical stability. This is an elegant, efficient formu-

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lation that y ields robust numerical solutions and physical insight of the VSC-based equipment for the steady-state and dynamic operating regimes.

The main scientific contributions of this thesis may be summarised as follows:

 Improved fundamental-frequency, positive-sequence representations of voltage-sourced converters for steady-state and dynamic simulations of power systems.

 A method to provide frequency regulation in low-inertia (island) AC grids using point-to-point VSC-HVDC links.

 Comprehensive steady-state and dynamic models of STATCOMs, back-to-back HVDC links, point-to-point HVDC links and multi-terminal HVDC systems bearing in mind their main operational characteristics.

 Development of a generalised frame-of-reference for the unified iterative solution of AC/DC networks. Both steady-state and time-domain simulations are covered.

1.2 List of published papers

Journal papers:

[1] Luis M. Castro and Enrique Acha, “A Unified Modeling Approach of Multi-Terminal VSC- HVDC L inks for Dynamic Simulations of Large-Scale Power Systems”, IEEE Transactions on Power Systems, Vol. 31, pp. 5051-5060, February 2016.

[2] Enrique Acha and Luis M. Castro, “A Generalized Frame of Reference for the Incorporation of Multi-Terminal VSC-HVDC Systems in Power Flow Solutions”, Electric Power Systems Re- search, Vol.136, pp. 415-424, July 2016.

[3] Luis M. Castro and Enrique Acha, “On the Provision of Frequency Regulation in Low Inertia AC Grids using VSC-HVDC links”, IEEE Transactions on Smart Grid, Vol. 7, pp. 2680-2690, November 2015.

[4] Luis M. Castro, Enrique Acha, C. R. Fuerte-Esquivel, “A novel VSC-HVDC Link Model for Dy- namic Power System Simulations”, Electric Power Systems Research, Vol. 126, pp. 111-120, May 2015.

Conference papers:

[5] Enrique Acha and Luis M. Castro, “Power Flow Solutions of AC/DC Micro-grid Structures”, 19^th Power Systems Computation Conference (PSCC’16), Genoa, Italy, 19-24 June 2016.

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5 The author of this thesis had the main responsibility on the development of the dynamic modelling framework of the two-terminal and multi-terminal HVDC links reported in [1], [3] and [4], where the writing of the papers was also carried out by the author. In these publications, Prof. Enrique Acha was responsible for ensuring the quality of the writing process, contributing actively with in- sightful observations which enabled the realisation of the overall dynamic framework relating to the simulation of HVDC systems. The latter reference was also supported by Prof. C. R. Fuerte- Esquivel who contributed with helpful suggestions which facilitated the incorporation of the control strategy of the HVDC link controlling the DC power flow; his very thoughtful comments helped improve the quality of the paper. Concerning references [2] and [5], Prof. Enrique Acha settled the path for the solution of multi-terminal HVDC systems for power-flow solutions being also responsible for writing up most of the sections included in these papers. The author participated actively in carrying out, developing and implementing in software the modelling framework by himself, something that gave a point of comparison of the obtained results; this was necessary since the power- flow solutions with multi-terminal HVDC systems are used as the starting point in the dynamic simulations of AC/DC power networks. In these two papers, the author wrote the sections relating to the reported study cases.

1.3 Thesis outline

The remainder of this thesis is organised as follows:

Chapter 2 presents the state-of-the-art concerning this topic of research, with emphasis on the early attempts to model VSC-based equipment for both steady-state and dynamic simulations of power systems. It argues the merits of this research work in advancing the modelling of the various power electronics devices put forward in this research, such as the STATCOM, back-to-back and point-to-point HVDC links and multi-terminal HVDC systems.

Chapter 3 addresses the steady-state, positive sequence models of the AC circuit of the STAT- COM, back-to-back and point-to-point HVDC links and multi-terminal HVDC systems. Each model is developed based on the power injections concept and solved within the context of a unified power flow formulation using the Newton-Raphson method, where general guidelines for the initial- isation of their state variables are provided. The solutions obtained with this frame-of-reference are useful for obtaining the steady-state conditions of the whole network’s models, i.e., generators, transmission lines, power transformers and the new VSC-based equipment. The convergent solution represents the equilibrium point with which the follow up dynamic simulations are carried out.

Chapter 4 presents a comprehensive dynamic model of the STATCOM, back-to-back and point-to- point HVDC links and multi-terminal HVDC systems aimed at fundamental-frequency, positive-

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sequence dynamic simulations of power systems. These models are developed from first princi- ples, encapsulating the key physical and operational characteristics of the equipment, i.e., each VSC unit is endowed with two degrees of freedom to provide AC voltage control and DC voltage or DC power regulation. The AC nodal voltage regulation is carried out with the amplitude modulation index, the DC voltage control ensures a stable converters operation. In the multi-terminal application, three types of dynamic control strategies are developed to match the specific operational requirements of each converter’s pairing AC s ub-network. In a similar way to the case of the power flows addressed in Chapter 2, the dynamic modelling and solution approach follows an efficient, unified methodology. The dynamic model of the VSC unit is used as the basic building block with which the various VSC-HVDC models are built. They may include only two VSCs for the case of back-to-back and point-to-point schemes or an arbitrary number of VSCs for cases of multi- terminal systems.

Chapter 5 illustrates the applicability of the newly developed models to carry out power flow solutions and time-domain simulations using popular test power networks. These networks range in size from a few nodes only to a 39-node network. More importantly, to demonstrate that the models yield realistic results concerning both the steady-state and dynamic operating regimes, compar- isons of the point-to-point and multi-terminal HVDC systems are carried out against switching- based models, giving a reasonable match between the two types of simulations. However, it is shown that the new models outperform the EMT-type models in terms of the computing time incurred in the simulations by approximately ten times, thus demonstrating their applicability within a context of practical power networks.

Chapter 6 gives the general conclusions of the thesis and provides suggestions for future research work.

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2 State-of-the-art of the Thesis

In contrast to earlier reactive power control devices, such as the Static Var Compensator (SVC), voltage source converters afford an excellent under voltage performance, enhancing voltage stability, thus increasing the power system transfer capability. The high performance of VSC stations stems from the fast action of the PWM-driven IGBT (Insulated Gate Bipolar Transistors) valves which enable the power converters to maintain a smooth voltage profile at its connecting node, even in the face of severe disturbances in the power grid. This is so because of its fast VAR s up- porting function by pure electronic processing of the phase-shifting of the voltage and current waveforms. Both, reduction of the converter switching losses (from 3% in earlier designs to less than 1% in actual converters) and larger operating power ranges up to 2 GW have become the driving force of their implementation on a global scale (ABB, 2014). As a result of its unabated progress, new converter topologies and control algorithms are being developed by commercial ven- dors in their factories and in research laboratories in Europe and elsewhere. The current drive is in multi-level converters aimed at lowering, even further, its powe r losses and improving the quality of the AC waveforms with little filtering. It may be argued that the current high level of penetration of power electronics in AC power grids is due to the flexibility and reliability afforded by present day VSC stations using IGBT switches (Gong, 2012; Bauer et al., 2014).

The leading developers of the VSC technology are ABB and Siemens; both use different commercial brands to refer to their own developments relating to VSC-based equipment, i.e., ABB uses the suffix Light^® and Siemens uses the suffix Plus^®. In both cases, each VSC unit is employed as a building platform to give rise to very important power system components within the family of FACTS devices; among these are the STATCOM, the back-to-back HVDC link, the point-to-point HVDC link, and recently, the concept of DC grids by means of multi-terminal HVDC links. All of them are key enablers for providing reactive power control (Schauder et al., 1997; Dodds et al., 2010), feeding of city centres with high power demand (Jacobson and Asplund, 2006), interconnecting two independent networks (Petersonn and Edris, 2001; Larsson et al., 2001), transmitting power through submarine cables (Ronström et al., 2007), tapping into onshore and offshore wind

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power plants (Robinson et al., 2010), harnessing photovoltaic resources into AC power grids (Kjaer et al., 2005), strengthening weak AC systems (Nnachi et al., 2013; Beccuti et al., 2014), voltage and power control in microgrids (Sao and Lehn, 2008; Chung et al., 2010; Rocabert et al., 2012; Eghtedarpour and Farjah, 2014), powering oil and gas rigs that lie in deep waters (Stendius and Jones, 2006; Gilje et al., 2009) and feeding into island networks with no generation of their own (Guo and Zhao, 2010; Zhang et al., 2011). It is in this context that some influential bodies, such as the pan-European electricity market, see as essential that an electricity grid in the North Sea be built (Trötscher et al., 2009). The available options point towards a multi-terminal HVDC system using VSC technology (Vrana et al., 2010). Owing to its great engineering complexity and large investment, a power grid of this nature requires a modular approach to its construction. It is an enterprise where several countries will be contributing technically and financially, just as in its operation (Hertem and Ghandhari, 2010). Plans are well advanced and it may be argued that its construction has already begun as part of the Bard offshore 1 wind park; an 80 5-MW wind turbine development lying 100 km off the German coast. It is argued in (Cole et al., 2011) that some of the advantages of having such an electricity grid in the North Sea would be the tapping of wind parks which might not be feasible since they would require their own cable connection to the s hore, as well as the efficient harvesting of fossil fuels in the North Sea.

2.1 Early attempts to modelling VSC-based equipment for power sys- tems simulations

An IGBT-based VSC coupled to an on-load tap-changing (OLTC) transformer is commonly referred to as a STATCOM whose converter is built as a two-level or a modular multi-level inverter operating on a constant DC voltage. It contains capacitors on its DC side whose sole function is to support and stabilise its DC voltage to enable the converter operation. As a result of its VAR generation/absorption by electronic processing of voltage and current waveforms, there is no need for additional capacitor banks and shunt reactors. Indeed, the power network sees the STATCOM as an electronic generator fulfilling the functions of a synchronous condenser but with no inertia of its own, this being the reason why this device is considered as a voltage source behind a reactance in a vast variety of simulation tools of power systems for both the steady-state and dynamic regimes.

As a result, the concept of a controllable voltage source behind a coupling impedance has been a popular modelling resource to represent the steady-state fundamental frequency operation of the STATCOM (Acha et al., 2005). This simple model might explain well the operation of the STAT- COM from the standpoint of the AC network but its usefulness is very much reduced when the requirement involves the assessment of variables relating to its DC bus. The situation is very much the same when looking at the dynamic regime where the standard approach has also been the use of a controllable voltage source (Cañizares 2000; Faisal et al., 2007; Barrios-Martinez et al., 2009;

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9 Shahgholian et al., 2011; Mahdavian and Shahgholian, 2011; Wang and Crow, 2011). All these modelling approaches use the power flowing into the equivalent voltage source to directly control the DC voltage magnitude – to a greater or lesser extent, the idea l voltage source is treated as the DC bus of the STATCOM. Following this idea, it was shown in (Jabbari et al., 2011) that the STATCOM may be treated as a controllable reactive c urrent source with a time delay response, one where its representation itself hinders the possibility, at best, of calculating the dynamic behaviour of the DC voltage.

Two voltage source converters linked on their corresponding DC side gives rise to multi-purpose applications of power converters. This scheme makes feasible the interconnection of two otherwise independent AC systems which may, or may not, be operating with different electrical frequencies;

it is said that this is an asynchronous interconnection of AC systems, whose configuration is commonly referred to as VSC-HVDC link, a well-established application in the arena of high-voltage, high-power electronics. Physically, both VSC stations can be positioned in the same place (referred to as back-to-back connection) or be separated by a DC cable of a certain length (referred to as point-to-point connection). In both cases, the valve switching of the IGBTs driven by PWM controls, at the two VSC stations comprising the high-voltage DC link, permits to regulate dynamically and in an independent manner, both the reactive power injection at either terminal of the AC system and the power flow through the DC link (Latorre et al., 2008). Overall this power electronic device resembles the interconnection of two STATCOMs connected through their DC buses, hence, not surprisingly, the early attempts to modelling VSC-HVDC systems for power systems simulations, resorted to modelling the two VSCs by idealised voltage sources (Zhang, 2004; Acha et al., 2005; Ruihua et al., 2005; Teeuwsen, 2009). Alternatively, the two converters comprising the VSC-HVDC link have also been represented by equivalent controlled current sources (Cole and Belmans, 2008; Cole and Belmans, 2011), where the currents injected into the AC systems are computed by the existing difference between the complex voltage of the VSC terminals and the AC system bus at which the two conveters of the VSC-HVDC link is embedded. Although it is advantageous to represent each VSC comprising the HVDC link as a controlled voltage source owing to its much reduced complexity, the natural trade-off, when resorting to this modelling approach, will be not having its internal variables readily available, i.e., the DC voltages or the modulation ratio of the converter stations. In an attempt to circumvent the problem of not having available key parameters such as the modulation indices and DC voltages, lossless VSC models, with different control strategies, aimed at modelling two-terminal VSC-based transmission systems for power flows were developed in (Watson and Arrillaga, 2007). Besides the reduced modelling complexity of the VSCs, the employed solution approach to solving power networks including HVDC links was sequential, one where the interconnected AC networks are solved using the Fast Decoupled power-flow method and the DC variables are computed separately through the Newton’s method, thus introducing frailty in the overall formulation, from the numerical standpoint. On the other hand, re- garding the time-domain solutions of VSC-HVDC links, the c oncept of dynamic average modelling

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10

has caught the attention of the power system community since it allows the modelling of VSC- HVDC systems in a detailed manner (Chiniforoosh et al., 2010), but with relatively reduced computing time comparing to the time incurred in highly-detailed HVDC models used in electromagnetic- transient (EMT) simulators. In the dynamic average modelling approach, the average value of the output voltage waveform is calculated at each switching interval, a value that changes dynamically depending on the value of the reference waveform. Each power converter comprising the HVDC link is represented by a three-phase controlled voltage source on the AC side and as a controlled current source on the DC side (Moustafa and Filizadeh, 2012). This approach may be time consuming when repetitive simulations studies are required in large-scale power system applications, for instance, in power system expansion planning and in operation planning. The solution time is always an important point to bear in mind and in the dynamic average modelling approach, increasing time steps is always a temptation but caution needs to be exercised when using this method since, as reported in (Moustafa and Filizadeh, 2012), the use of large integration time steps, during the numerical solution of the model, may affect the accuracy of the results.

VSC-HVDC systems are designed to serve in a wide range of power systems applications. Two interconnected AC networks by means of an HVDC link exhibit a natural decoupling in terms of both voltage and frequency. This type of interconnections is primarily aimed at preventing the ex- cursions of oscillations between AC systems, for instance, between a stronger and a weaker AC network. The back-to-back HVDC system linking the USA and Mexican power systems in the Ea- gle Pass substation and the submarine HVDC link interconnecting the power grids of Finland and Estonia are two examples of this application of the VSC-HVDC technology (Larsson et al., 2001;

Ronström et al., 2007). Nevertheless, there are other applications where it is desirable to exert the influence of the strong network upon the weak network through the DC link. Such a situation would arise when power imbalances occur in an AC power network with little or no inertia and fed by a VSC-HVDC link (Guo and Zhao, 2010; Zhang et al., 2011). It should be noticed that in a power network with no inertia, even a very small power imbalance would lead the frequency to experi- ence large rises or drops, depending on the nature of the power imbalance. Furthermore, all AC networks contain a degree of frequency-sensitive loads and VSCs do not possess the ability to strengthen, on their own, the inertial response or to aid the primary frequency control of AC networks. This is a problem demanding research attention because AC networks with poor inertia are likely to become more common (Callavik et al., 2012). Preliminary research progress in the area of frequency support, using HVDC systems, in electrical networks with very low inertia has been made in various fronts, using a range of equipment models and control schemes embedded in software with which simulation studies are carried out (Guo and Zhao, 2010; Zhang et al., 2011;

Nnachi et al., 2013; Beccuti et al., 2014). For instance, microgrids fed by HVDC links have been a matter of research paying special attention to grid-forming, grid-feeding and grid-supporting topologies (Rocabert et al., 2012). The power control performance in AC microgrids, with their strong impact on frequency, are receiving far greater research attention because these types of grids are

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11 likely to become cost-effective solutions for the interconnection of distributed generators in power systems (Sao and Lehn, 2008; Chung et al., 2010; Eghtedarpour and Farjah, 2014). In this particular application of the VSC-HVDC link, the under-representation of the converters as ideal voltage sources may cause, as in the case of the HVDC link interconnecting two power systems pos- sessing each frequency-regulation equipment, the relevant phenomena that occur in the HVDC link not to be captured (Ruihua et al., 2005; Teeuwsen, 2009). As an attempt to overcome the underrepresented VSCs, switching-based models of the VSC-HVDC link together with three-phase AC power networks have been widely used to investigate the very timely subject of frequency support in low-inertia electrical networks (Guo and Zhao, 2010). However, the simulation studies associated with this approach are time consuming and therefore the AC networks used in such studies have been very contrived and, hence, poorly represented in terms of their actual size. To ac- celerate the solutions, the so-called dynamic average models have come into play since they enable a detailed representation of the power converters with moderate solution times (Chiniforoosh et al., 2010). As stated earlier, this solution may still incur large computing times, especially in applications of HVDCs providing frequency support to low-inertia grids, one that implicates mid-term to long-term dynamic simulation studies (simulation times of 5 seconds and more), since these are the time scales involved in most frequency oscillations phenomena present in power systems.

Point-to-point HVDC designs have advanced in the direction of multi-terminal VSC-HVDC systems which are currently at the frontier of VSC-based technology developments. The super DC grid which is being constructed in the North Sea is perhaps the most ambitious engineering project that will be built over the next few years (Vrana et al., 2010; Hertem and Ghandhari, 2010; Orths et al., 2012). Its operation will require skilful electrical engineers with suitable tools to deal with an array of technical problems that may emerge from its daily operation. Research interest in multi- terminal VSC-HVDC systems is relatively new; power systems’ application tools are being upgrad- ed to incorporate multi-terminal VSC-HVDC models to the conventional formulations of power flows and transient stability in order to conduct system-wide studies. For instance, to obtain the steady-state operating point of electrical networks, power-flow studies are carried out using sequential and quasi-unified solutions of the AC/DC networks formed by multi-terminal HVDC systems (Chen et al., 2005; Beerten et al., 2010; Ye et al., 2011; Baradar and Ghandhari, 2013, Wang and Barnes, 2014). In general terms, both methods compute the state variables of the AC sub- networks through a conventional power-flow algorithm, whilst a sub-problem is formulated for calculating the state variables of the DC network, something that is carried out at each iteration of the power-flow solution, until a convergence tolerance is reached. This sequential approach is attrac- tive because it is straightforward to implement it in existing power flow programs, but caution has to be exercised because it might yield, at best, a non-quadratic convergence in Newton-based power-flow algorithms (Smed et al., 1991; Fuerte-Esquivel and Acha, 1997). In the context of dynamic studies of multi-terminal HVDC systems, a similar situation arises. Conducting holistic dynamic analyses of resultant AC/DC networks is not an easy task due to the size a nd complexity of

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12

such large systems, although creditable efforts have been made (Cole et al., 2010; Chaudhuri et al., 2011; Peralta et al., 2012; Adam et al., 2013; Shen et al., 2014; Van der Meer et al., 2015). The simulation of innumerable scenarios is an inevitable requirement in power system dynamic analysis, hence, multi-terminal HVDC models which depart from a detailed EMT-type representation have been recently proposed aiming at reducing the computing time of the s imulations (Peralta et al., 2012; Van der Meer et al., 2015), this being a major concern in practical networks. Not surprisingly, existing approaches use contrived, idealised voltage or current sources to represent each VSC station comprising the multi-terminal scheme, a fact that leads to executing additional calcula- tions to adjust the power flow control in selected VSC stations (Adam et al., 2013) or solving the AC sub-networks and the DC network in a s eparated manner (Cole et al., 2010), thus introducing frailty in the formulation.

2.2 Advances in the state-of-the-art modelling of VSC-based equipment for power systems simulations

The field of high-voltage, high-power electronics is evolving rapidly, in a certain sense, thanks to the incentives and encouragements by governments and civil associations to employ cleaner en- ergies as a platform for the development of future societies. Such is the case of the energy- harvesting and integration of a great amount of renewable resources into the power systems, or simply, the power exchange between neighbouring countries to avoid the use of fossil fuel-based power plants. Although this has brought about a relatively higher penetration of VSC-based equipment, there has been limited progress in terms of developing realistic models for system-wide studies. It may be argued that representative models of the network have traditionally been the cornerstone of planners and analysts of power systems because counting on suitable models and simulation tools enable them to anticipate operational problems that may emerge from the daily operation of power networks. In this sense, models of STATCOMs, back-to-back and point-to-point VSC-HVDC links and multi-terminal VSC-HVDC systems aimed at utility-scale electrical systems are also of major concern for power systems analysts.

The use of these VSC-based power devices has been on the increase in recent years. Paradox- ically, the standard practice in their modelling approach has not fundamentally changed in the sense that reduced models are being used in order to facilitate, for instance, their implementation in existing power system s imulation tools. On the contrary, if highly detailed models are employed, such as those found in EMT-type simulation packages, then excessive simulation times may be incurred; this is a fact that would impair their applicability in large power networks. Overall, this summarises the motivation of this thesis where the emphasis is placed on the investigation of a novel VSC model which may be used, with due extensions, as a modelling platform to represent

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13 the STATCOM and any possible configuration of VSC-HVDC links including the so-called multi- terminal arrangement with arbitrary DC network topologies.

The whole family of models of VSC-based devices, developed in this thesis, represents a paradigm shift in the way voltage source converters are modelled and in the manner they are solved through a unified frame-of-reference for both power flow simulations and positive-sequence dynamic simulations. At first, the developed VSC itself contrasts with those in current use, departing from the controllable voltage source concept so far used to represent the converter. It rather uses a compound of a phase-shifting transformer and an equivalent shunt admittance to represent the phase-shifting and scaling nature of the PWM control, respectively, resulting in a far superior modelling flexibility when it comes to representing the AC and DC s ides of the converter. The AC terminal of the power converter combines quite easily with the model of the interfacing load-tap changing transformer to make up the full VSC station found in STATCOMs or VSC-HVDC links.

Given that the DC bus of the converter is explicitly represented, any basic circuit element may be attached to it, something that is particularly advantageous when setting up multi-terminal schemes where one or various DC transmission lines may connect to the DC node of the converter. All these facts result in a straightforward computation of the internal state variables of the VSCs, including the DC circuit, facilitating also the inclusion of conduction losses and switching losses.

Building up on this VSC model, carrying out the implementation of back-to-back and point-to- point VSC-HVDC models is not a difficult task; it only requires bearing in mind the operating fea- tures of this equipment. The four degrees of freedom found in actual VSC-HVDC installations, characterised by having simultaneous voltage support at its two AC terminals, DC voltage control at the inverter converter and regulated DC power at the rectifier converter, can be inherited to these models subjected to their particular application, i.e., to provide a constant powe r flow regulation through the DC link or to exert frequency regulation to low-inertia AC networks. Despite the pursued application, the grid-oriented VSC-HVDC models, developed in this thesis, encapsulate the essential steady-state and dynamic characteristics of the AC networks, the load-tap c hanging transformers, the power electronic converters and the DC transmission line, where the ensuing dynamic model is solved at only a fraction of the time than that required by highly-detailed, switching-based models. For instance, the four degrees of freedom together with a communication chan- nel between the two converter stations are exploited to investigate the viability of exerting frequency control in low-inertia AC grids fed by a VSC-HVDC link, whose application lies in the realm of the mid-term to long-term power system dynamics involving several seconds of simulation, with the inverter station seen to act as a power electronic source capable of controlling the voltage magnitude and phase angle at its AC bus, hence, effectively acting as the slack bus of the AC grid with near-zero inertia.

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14

With regards to the modelling of multi-terminal schemes, any converter might play the role of either a rectifier or inverter; this is so because the power may flow in a bidirectional fashion, i.e., from the DC grid to the AC network or vice versa, according to the prevailing operating conditions. In this context, the enhanced representation of the VSC model permits to make due provisions for three types of converters and their control strategies: the slack VSC whose aim is to control the DC voltage, the scheduled-power VSC that c ontrols the power transfer and the passive VSC which is connected to AC networks that have no load/frequency control equipment. This gives rise to a comprehensive formulation, not found in other approaches, which is s traightforwardly expandable to build up a model that comprises a number of VSC units which is commensurate with the number of AC sub-networks. More importantly, any type of AC/DC system, which is represented within this framework, will not omit the calculation of important variables of the VSCs, as is the case of the modulation ratio in other methods (Cole et al., 2010; Adam et al., 2013; Shen et al., 2014).

The overall steady-state and dynamic models of the new range of power electronic devices presented in this thesis are developed in an all-encompassing frame-of-reference based on the concept of power injections, where the non-linear algebraic equations of the power system, synchronous generators, STATCOMs, back-to-back and point-to-point VSC-HVDC links and multi-terminal VSC-HVDC links including its DC network are linearised around a base operating point and com- bined together with the discretised differential equations arising from the controls of the various equipment being modelled, for unified iterative solutions using the Newton-Raphson method. One such iterative solution is valid for a given point in time and its rate of convergence is quadratic.

Moreover, the differential equations are discretised using the implicit trapezoidal method for enhanced numerical stability. This is a time-domain solution where the emphasis is placed on the dynamic models of the VSC-based equipment which is comprehensive, quite elegant and yields physical insight. It should be remarked that the new models put forward are not switching-based models (EMT-type models), such as those used in commercial packages as PSCAD and SIM- ULINK where the PWM pattern is fully emulated together with the switching action of each converter’s IGBT valve. Rather, this model follows the standard way of representing electrical equipment and their controls in large-scale electrical power system modelling and simulations (Stagg and El-Abiad, 1968) – they are said to be lumped-type models. These models take the approach of representing one full period of the fundamental frequency waveform by a phasor corresponding to the base waveform’s frequency. Lumped-type models are used in a number of industry-grade commercial packages, such as PowerWorld and PSS^®E. In this formulation, the Jacobian matrix of the entire system becomes available, at any point in time, opening a window of applications for many other power systems studies, such as eigenvalue tracking, dynamic sensitivity analysis and evaluation of dynamic stability indices, although these analyses are not addressed in this thesis.

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3 Modelling of VSC-based equipment for power flows

The power electronics equipment that emerged from the FACTS initiative (Hingorani and Gyugyi, 1999) has a common purpose: to alleviate one or more operational problems at key locations of the power grid. A case in point is the use of VSC-based FACTS controllers such as the STATCOM and VSC-HVDC systems. This technology is designed to regulate the reactive power at its point of connection with the grid, in response to both fast and slow network voltage variations. Its good performance arises from the fast action of the PWM-driven IGBT valves which enable the VSCs to maintain a smooth voltage profile at its connecting node, even in the face of severe disturbances in the power grid.

This chapter presents enhanced models of the STATCOM, back-to-back, pointo-to-point and multi-terminal VSC-HVDC links suitable for the steady-state operating regime of power grids. In the new models, the VSC itself is represented as a phase-shifting transformer; one where its primary and secondary sides yield quite naturally to the AC and DC sides of the power converter, respectively. Hence, the VSC model and by extension those of the STATCOM, two-terminal and multi-terminal VSC-HVDC links, take into account in aggregated form, the phase-shifting and scaling nature of the PWM control. Furthermore, the AC terminal of the VSC combines quite easily with the model of the interfacing load tap-changing transformer whereas the explicit representation of its DC side permits to connect, if required, to DC transmission lines or any other basic power system elements to its DC bus. Therefore, the modelling approach adopted for the STATCOM and VSC-HVDC systems is incremental in nature, whose frame-of-reference accomodates quite naturally any number of AC/DC networks generated by an arbitrary number of VSC converters.

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3.1 Voltage source converter model

The combination of a voltage source converter and a load-tap changing transformer, as shown in Figure 3.1(a), is usually termed STATCOM. The AC terminal of the VSC is connected to the secondary winding of the OLTC transformer which plays the role of an interface between the power converter and the AC power grid, adding a further degree of voltage magnitude controllability. Physically, the VSC is built as a two-level or a multi-level converter that uses an array of self-commutating power electronic switches driven by PWM control. On its DC side, the VSC employs a capacitor bank of relatively small rating whose sole function is to support and stabilise the voltage at its DC bus to enable the converter operation. The converter keeps the capacitor charged to the required voltage level by making its output voltage lag the AC system voltage by a small angle (Hingorani and Gyugyi, 1999). The DC capacitor of value C_dc is shown quite prominently in Figure 3.1(a). It should be mentioned that C_dc is not responsible per se for the actual VAR generation process and certainly not at all for the VAR absorption process. Instead the VAR generation/absorption process is carried out by the PWM control to satisfy operating requirements. Such an electronic processing of the voltage and current waveforms may be well characterised, from the fundamental frequency representation standpoint, by an equivalent susceptance which can be either capacitive or inductive to conform to operating conditions.

Figure 3.1 (a)STATCOM schematic representation; (b) VSC steady-state equivalent circuit The VSC operation at the fundamental frequency may be modelled by means of electric circuit components, as shown in Figure 3.1(b). The electronic processing of the voltage and current waveforms of the VSC is well synthesised by a notional variable susceptance which is resposible for the whole reactive power production in the valve set of the converter. Furthermore, the ideal phase-shifting transformer with complex taps is the key component that enables a dissociation, from the voltage angle standpoint, between the AC voltage V₁ and the DC voltage E_dc, resulting in the element that provides the interface AC and DC circuits of the VSC, which follows the ensuing basic voltage relationship (Acha and Kazemtabrizi, 2013).

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17

1 2

j

k m E ea dc ^



V (3.1)

where the complex voltage V₁ is the voltage relative to the system phase reference, the tap magnitude m_a of the ideal phase-shifting transformer corresponds to the VSC’s amplitude modulation coefficient where the following relationship holds for a two-level, three-phase VSC: k2 ³8, the angle  is the phase angle of voltage V₁, and E_dcis the DC bus voltage which is a real scalar.

The steady-state VSC model comprises an ideal phase-shifting transformer with complex taps connected in series with an impedance, an equivalent variable shunt susceptance B_eq placed on the right-hand side of the transformer and a resistor on its DC side, as seen on Figure 3.1(b). The series reactance X represents the VSC’s interface magnetics whereas the series resistor R is associated to the ohmic losses which are proportional to the AC terminal current squared. The switching loss model corresponds to a constant resistance (conductance G₀) ,which under the presence of constant DC voltage and constant load current, would yield constant power loss for a given switching frequency of the PWM converter. Admittedly, the constant resistance characteristic may be inaccurate because although the DC voltage is kept essentially constant, the load current will vary according to the prevailing operating condition. Hence, the shunt resistor (with a conductance value of G_sw) produces active power losses to account for the switching action of the PWM converter. This conductance is calculated according to the existing operating conditions and ensures that the switching losses are scaled by the quadratic ratio of the actual terminal current magnitude I_v to the nominal currentI_nom,

 

²

0 /

sw v nom

G G I I (3.2)

The phase-shifting transformer making up the VSC model plays a crucial role in describing the converter operation; it decouples, angle-wise, the circuits connected at both ends of the transformer. This has the immediate consequence that the AC and DC circuits brought about by the inclusion of one or more VSCs in the power network c an be s olved by using conventional AC power system applications, such as the AC power flows . It is worth mentioning that the voltage E_dc is directly linked to the AC voltage V₁ where only the phase-shifting transformer lies in between. Ar- guably, both voltages may be seen to be linked to the same connection point, and hence, to the same AC system. This fact is akin to that resulting from the steady-state modelling of idealised power transformers, in per-unit basis, employed in conventional power flow theory where, in a notional manner, the high-voltage and low-voltage sides are separated by a mere connection point.

However, in the case of the VSC, its idea l phase-shifting transformer not only links two different voltage levels, but also, these two voltages may be seen to correspond to different current sys-

Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power Networks

Luis Miguel Castro González

Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power Networks

Julkaisu 1445 • Publication 1445

Tampere 2016

Tampereen teknillinen yliopisto. Julkaisu 1445 Tampere University of Technology. Publication 1445

Luis Miguel Castro González

Modelling of Multi-terminal VSC-HVDC Links for Power Flows and Dynamic Simulations of AC/DC Power

Networks

Tampereen teknillinen yliopisto - Tampere University of Technology

Tampere 2016

ISBN 978-952-15-3866-7 (printed)

ISBN 978-952-15-3869-8 (PDF)

ISSN 1459-2045

Abstract

Preface

Contents

List of Figures

List of Tables

List of Symbols and Abbreviations

1 Introduction

1.1 Objectives and scientific contribution

1.2 List of published papers

1.3 Thesis outline

2 State-of-the-art of the Thesis

2.1 Early attempts to modelling VSC-based equipment for power sys- tems simulations

2.2 Advances in the state-of-the-art modelling of VSC-based equipment for power systems simulations

3 Modelling of VSC-based equipment for power flows

3.1 Voltage source converter model

 